# Re: Question about PA and 1p

```On 1/17/2012 5:08 AM, Bruno Marchal wrote:
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`SNIP`
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- I disagree that set theory might be more primitive than arithmetic. Why? First because arithmetic has been proved more primitive than set theory, and less primitive than logic. With logic we cannot define numbers. with set, we can define numbers, even all of them (N, Z, Q, ... octonions, etc.). The natural numbers are often defined by the von Neuman finite ordinal:
```
0 = { }
1 = {{{}}}  = {0}
2 = {{}, {{}} } = {0, 1}
3 = {{}, {{}},{{}}},,{{} ,{{}} } } = {0, 1, 2}
...
n = {0, 1, 2, ..., n-1}
etc.

```
And you can define addition by the disjoint union cardinal, and multiplication by the cardinal of cartesian product, and then, you can *prove* the laws of addition and the laws of multiplication. With arithmetic you cannot recover any axioms of set theory, except for the hereditarily finite sets.
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I am confused. It seems to me that you are admitting that sets are more primitive than Arithmetic since what you wrote here is a demonstration of how numbers supervene on set theoretic operations. The fact that we can define the natural numbers via the von Neuman finite ordinals is the equivalent of claiming that the natural numbers emerge from the von Neuman finite ordinals (up to isomorphism!), so I am confused by what you are claiming here! But whichever is the most primitive, it is not more primitive than the neutral foundation of existence in itself.
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- As I said, I don't take the word "Existence" as a theory. I have no clue what you mean by that. I was asking for a theory. You say that by taking (N, +, *) as a primitive structure, I am no more neutral monist, due to the use of + and *. This is not correct. It would make neutral monism empty. We alway need ontological terms (here 0, s(0) etc.) and laws relating those terms (here addition and multiplication).
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No, I am not making Existence as a theory, it is merely a postulate of my overall theory (if you can call what I have been discussing a "theory"). I am using the notion ofExistence <http://books.google.com/books?id=VttF6CuC-cQC&pg=PT170&lpg=PT170&dq=existence+Objectivist+epistemology&source=bl&ots=d2ZAMFVpJM&sig=CLaMS0Y9kVnB6UfbgwUsuCG3wsU&hl=en&sa=X&ei=vsMVT7WNM8nWtgfEvaTRAw&sqi=2&ved=0CFUQ6AEwBg#v=onepage&q=existence%20Objectivist%20epistemology&f=false> as it is defined in Objectivist Epistemology. For example, as explained in this video lecture: http://www.youtube.com/watch?v=GfOS7xfxezA&feature=player_embedded Neutral monist takes is empty in the sense that it shows the coherent implication that the most basic ontological level cannot be considered to have some definite set of properties to the exclusion of others.
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- All you arguments with the term "physical" are going through in arithmetic, given that you agree that "physical" is not primitive. For example, the physical world is not required to make sense of what is a universal machine. It is required for human chatting on the net, but such a physical world is provided by arithmetic. Including concurrency.
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WE simply might have to agree to disagree.

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- I don't do philosophy. I offer you a technical result only. I still don't know if you grasped it, or if you have any problem with it.
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You result has deep philosophical implications and as a student of philosophy I am very interested in it.
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If you agree to assume that the brain works like a material machine, then arithmetic is enough and more than arithmetic is necessarily useless: it can only make the mind body problem unecessarily more complex. Primitive matter (time, space) becomes like invisible horse. Not epiphenomena, but epinomena.
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Again, we have to agree to disagree on this. The necessity of physical implementation cannot be dismissed otherwise the scientific method itself is empty and useless. Without the definiteness that the physical world offers us is accepted there can be only idle speculation, we saw this kind of thinking in the Scholastics <http://en.wikipedia.org/wiki/Scholasticism> and know well how that was such a terrible waste of time. So why are you advocating a return to that? Ideal monism was pushed hard by Bishop Berkeley and failed back in the 18th century, its flaw - that the material world becomes causally ineffective epiphenomena - is not solved by your result, it is only more explicitly shown. You seem to think that it is a virtue. No, sorry, it is not a virtue for the simple reason that it makes the falsification of the theory impossible thus rendering it useless as an explanation. My alternative hypothesis has the chance of being falsified as it predicts that the physical worlds actually observe must be representable as Boolean algebras (up to isomorphisms).
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Onward!

Stephen

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